T. F. Thonn and R. M. A. Azzam, "Three-reflection halfwave and quarterwave retarders using dielectric-coated metallic mirrors," Appl. Opt. 23, 2752-2759 (1984)
A design procedure is described to determine the thicknesses of single-layer coatings of a given dielectric on a given metallic substrate so that a specified net phase retardance (and/or a net relative amplitude attenuation) between the p and s polarizations is achieved after three reflections from a symmetrical arrangement of three mirrors that maintain collinearity of the input and output beams. Examples are presented of halfwave and quarterwave retarders (HWR and QWR) that use a ZnS–Ag film–substrate system at the CO2-laser wavelength λ = 10.6 μm. The equal net reflectances for the p and s polarizations are computed and found to be high (above 90%) for most designs. Sensitivity of the designs (deviation of the magnitude and phase of the ratio of net complex p and s reflection coefficients from design specifications) to small film-thickness and angle-of-incidence errors is examined, and useful operation over a small wavelength range (10–11 μm) is demonstrated.
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Summary of Design Results for ZnS–Ag Three-Reflection Halfwave Retarders (HWR: |ρn| = 1, Δn = δpn − δsn = ±180°) at Two Angles of Incidence and for Wavelength λ = 10.6 μma
ϕ1
ζ1
ζ2
Dϕ1
Dϕ2
ℛνn
δpn
δsn
80 A
0.31634
0.40638
2.6941
2.6644
0.9672
−96.965
83.035
B
0.39404
0.29610
2.6941
2.6644
0.9689
−89.782
90.218
C
0.59110
0.67256
2.6941
2.6644
0.9689
91.178
−88.822
D
0.65557
0.58024
2.6941
2.6644
0.9675
97.402
−82.598
85 A
0.11938
0.44629
2.7020
2.6941
0.9557
−67.599
112.401
B
0.43494
0.07680
2.7020
2.6941
0.9627
−58.252
121.748
C
0.54840
0.90090
2.7020
2.6941
0.9629
58.296
−121.704
D
0.85867
0.53708
2.7020
2.6941
0.9560
67.523
−112.477
ϕ1 is the angle of incidence in degrees on mirrors M1 and M3; (ζ1,ζ2) is the normalized film-thickness solution pair; Dϕi is the film-thickness period in microns at ϕi,i = 1,2; ℛνn is the net reflectance of the retarder for both polarizations (ν = p,s); and δpn,δsn are the net phase shifts for the p and s polarizations in degrees. The refractive indices of ZnS and AG at 10.6 μm are taken to be 2.2 and 9.5–j73, respectively.
Table II
Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1 = Δd2 = ±1 nm), (II) Angle-of-Incidence Errors (Δϕ1 = Δϕ2 = Δϕ3 = ±0.5°) to the HWR Designs of Table I
Error type
ϕ1 (deg)
Magnitude error (|ρn| − 1) × 104
Phase error (Δn ∓ 180°) (deg)
−
+
−
+
I
80 A
−1.010
1.018
−0.0613
0.0633
B
−1.017
1.027
−0.0577
0.0597
C
1.007
−0.999
−0.0597
0.0578
D
0.990
−0.985
−0.0626
0.0607
85 A
−3.345
3.380
−0.0865
0.0927
B
−2.536
2.586
−0.0714
0.0766
C
2.567
−2.525
−0.0761
0.0713
D
3.346
−3.310
−0.0921
0.0859
II
80 A
−2.061
2.088
6.0568
−6.1141
B
1.385
−1.430
6.1212
−6.1799
C
1.188
−1.230
−6.1639
6.2222
D
−1.777
1.797
−6.0734
6.1299
85 A
−6.922
8.039
12.1470
−12.5933
B
15.510
−15.953
8.7586
−9.0116
C
15.306
−15.745
−8.8312
9.0848
D
−6.934
8.030
−12.1470
12.5889
Table III
Summary of Design Results for ZnS–Ag Three-Reflection Quarterwave Retarders with p Fast Axis (QWR: |ρn| = 1, Δn = δpn − δsn = 90°) at Three Angles of Incidence and for Wavelength λ = 10.6 μma
ϕ1
ζ1
ζ2
Dϕ1
Dϕ2
ℛνn
δpn
δsn
52 A
0.37716
0.68985
2.5751
2.4238
0.9445
178.381
88.381
B
0.37578
0.60396
2.5751
2.4238
0.9368
−143.182
126.818
75 A
0.68770
0.55294
2.6814
2.6207
0.9541
55.070
−34.930
B
0.59026
0.77066
2.6814
2.6207
0.9617
22.095
−67.905
85 A
0.91588
0.51795
2.7020
2.6941
0.9188
5.903
−84.097
B
0.54189
0.01364
2.7020
2.6941
0.9541
−28.401
−118.401
C
0.04588
0.51154
2.7020
2.6941
0.9001
17.318
−72.682
D
0.01175
0.48696
2.7020
2.6941
0.8578
108.005
18.005
E
0.38691
0.33921
2.7020
2.6941
0.9831
−138.976
131.024
F
0.34169
0.38882
2.7020
2.6941
0.9830
−139.512
130.488
See footnote of Table I for explanation of notation.
Table IV
Magnitude and Phase Errors Caused by Introducing(I) Film-Thickness Errors (Δd1 = Δd2 = ±1 nm), (II) Angle-of-incidence Errors (Δδ1 = Δϕ2 = Δϕ3 = ±0.5°) to the QWR Designs of Table III
Error type
ϕ1 (deg)
Magnitude error (|ρn| − 1) × 104
Phase Error (Δn − 90°) (deg)
−
+
−
+
I
52 A
−0.603
0.608
−0.0537
0.0550
B
−0.573
0.576
−0.0546
0.0557
75 A
1.072
−1.067
−0.0817
0.0797
B
1.101
−1.092
−0.0682
0.0660
85 A
7.479
−7.417
−0.1746
0.1609
B
2.988
−2.916
−0.0943
0.0884
C
3.982
−3.949
−0.2133
0.2046
D
−5.518
5.314
−0.3010
0.3106
E
−0.575
0.579
−0.0330
0.0341
F
−0.571
0.575
−0.0333
0.0344
II
52 A
−0.093
0.094
1.9350
−1.9622
B
−0.206
0.213
1.9246
−1.9512
75 A
−2.734
2.850
−5.1858
5.2592
B
1.402
−1.483
−5.0627
5.1303
85 A
−4.169
7.389
−14.0402
15.2408
B
24.356
−26.014
−5.7013
5.7001
C
0.813
2.351
8.0346
−9.1883
D
20.245
−25.010
7.1403
−8.4625
E
0.842
−0.858
6.6376
−6.6682
F
−1.007
1.000
6.5501
−6.5804
Table V
Summary of Design Results for ZnS–Ag Three-Reflection Quarterwave Retarders with s Fast Axis (QWR: |ρn| = 1, Δn = δpn − δsn = −90°) at Three Angles of Incidence and at Wavelength λ = 10.6 μm a
ϕ1
ζ1
ζ2
Dϕ1
Dϕ2
ℛνn
δpn
δsn
52 A
0.60610
0.01252
2.5751
2.4238
0.9520
−119.719
−29.719
B
0.61135
0.47032
2.5751
2.4238
0.9283
74.730
164.730
75 A
0.28732
0.43199
2.6814
2.6207
0.9538
−55.083
34.917
B
0.39292
0.19710
2.6814
2.6207
0.9620
−20.116
69.884
85 A
0.06262
0.46531
2.7020
2.6941
0.9185
−6.017
83.983
B
0.44122
0.96412
2.7020
2.6941
0.9542
28.667
118.667
C
0.93216
0.47129
2.7020
2.6941
0.9010
−16.303
73.697
D
0.96696
0.49627
2.7020
2.6941
0.8580
−108.245
−18.245
E
0.61142
0.61570
2.7020
2.6941
0.9832
139.889
−130.111
F
0.61542
0.61130
2.7020
2.6941
0.9832
139.940
−130.060
See footnote of Table I for explanation of notation.
Table VI
Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1 = Δd2 = ±1 nm),(II) Angle-of-Incidence Errors (Δϕ1 = Δϕ2 = Δϕ3 = ±0.5°) to the QWR Designs of Table V
Error type
ϕ1 (deg)
Magnitude error (|ρn| − 1) × 104
Phase error [Δn − (−90°)] (deg)
−
+
−
+
I
52 A
0.600
−0.598
−0.0537
0.0525
B
0.536
−0.534
−0.0561
0.0550
75 A
−1.069
1.070
−0.0802
0.0823
B
−1.086
1.097
−0.0656
0.0677
85 A
−7.450
7.509
−0.1615
0.1754
B
−2.898
2.965
−0.0881
0.0940
C
−3.931
3.968
−0.2024
0.2112
D
5.385
−5.588
−0.3104
0.3006
E
0.547
−0.543
−0.0339
0.0328
F
0.547
−0.543
−0.0339
0.0329
II
52 A
−0.087
0.087
−2.0732
2.1074
B
−0.497
0.519
−2.0382
2.0706
75 A
−2.882
3.002
5.1651
−5.2389
B
1.468
−1.553
4.9799
−5.0454
85 A
−4.030
7.251
14.0157
−15.2209
B
24.287
−25.924
5.5888
−5.5821
C
0.643
2.541
−8.0727
9.2161
D
20.129
−24.842
−7.1972
8.5240
E
0.034
−0.044
−6.6091
6.6386
F
−0.139
0.128
−6.5987
6.6281
Tables (6)
Table I
Summary of Design Results for ZnS–Ag Three-Reflection Halfwave Retarders (HWR: |ρn| = 1, Δn = δpn − δsn = ±180°) at Two Angles of Incidence and for Wavelength λ = 10.6 μma
ϕ1
ζ1
ζ2
Dϕ1
Dϕ2
ℛνn
δpn
δsn
80 A
0.31634
0.40638
2.6941
2.6644
0.9672
−96.965
83.035
B
0.39404
0.29610
2.6941
2.6644
0.9689
−89.782
90.218
C
0.59110
0.67256
2.6941
2.6644
0.9689
91.178
−88.822
D
0.65557
0.58024
2.6941
2.6644
0.9675
97.402
−82.598
85 A
0.11938
0.44629
2.7020
2.6941
0.9557
−67.599
112.401
B
0.43494
0.07680
2.7020
2.6941
0.9627
−58.252
121.748
C
0.54840
0.90090
2.7020
2.6941
0.9629
58.296
−121.704
D
0.85867
0.53708
2.7020
2.6941
0.9560
67.523
−112.477
ϕ1 is the angle of incidence in degrees on mirrors M1 and M3; (ζ1,ζ2) is the normalized film-thickness solution pair; Dϕi is the film-thickness period in microns at ϕi,i = 1,2; ℛνn is the net reflectance of the retarder for both polarizations (ν = p,s); and δpn,δsn are the net phase shifts for the p and s polarizations in degrees. The refractive indices of ZnS and AG at 10.6 μm are taken to be 2.2 and 9.5–j73, respectively.
Table II
Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1 = Δd2 = ±1 nm), (II) Angle-of-Incidence Errors (Δϕ1 = Δϕ2 = Δϕ3 = ±0.5°) to the HWR Designs of Table I
Error type
ϕ1 (deg)
Magnitude error (|ρn| − 1) × 104
Phase error (Δn ∓ 180°) (deg)
−
+
−
+
I
80 A
−1.010
1.018
−0.0613
0.0633
B
−1.017
1.027
−0.0577
0.0597
C
1.007
−0.999
−0.0597
0.0578
D
0.990
−0.985
−0.0626
0.0607
85 A
−3.345
3.380
−0.0865
0.0927
B
−2.536
2.586
−0.0714
0.0766
C
2.567
−2.525
−0.0761
0.0713
D
3.346
−3.310
−0.0921
0.0859
II
80 A
−2.061
2.088
6.0568
−6.1141
B
1.385
−1.430
6.1212
−6.1799
C
1.188
−1.230
−6.1639
6.2222
D
−1.777
1.797
−6.0734
6.1299
85 A
−6.922
8.039
12.1470
−12.5933
B
15.510
−15.953
8.7586
−9.0116
C
15.306
−15.745
−8.8312
9.0848
D
−6.934
8.030
−12.1470
12.5889
Table III
Summary of Design Results for ZnS–Ag Three-Reflection Quarterwave Retarders with p Fast Axis (QWR: |ρn| = 1, Δn = δpn − δsn = 90°) at Three Angles of Incidence and for Wavelength λ = 10.6 μma
ϕ1
ζ1
ζ2
Dϕ1
Dϕ2
ℛνn
δpn
δsn
52 A
0.37716
0.68985
2.5751
2.4238
0.9445
178.381
88.381
B
0.37578
0.60396
2.5751
2.4238
0.9368
−143.182
126.818
75 A
0.68770
0.55294
2.6814
2.6207
0.9541
55.070
−34.930
B
0.59026
0.77066
2.6814
2.6207
0.9617
22.095
−67.905
85 A
0.91588
0.51795
2.7020
2.6941
0.9188
5.903
−84.097
B
0.54189
0.01364
2.7020
2.6941
0.9541
−28.401
−118.401
C
0.04588
0.51154
2.7020
2.6941
0.9001
17.318
−72.682
D
0.01175
0.48696
2.7020
2.6941
0.8578
108.005
18.005
E
0.38691
0.33921
2.7020
2.6941
0.9831
−138.976
131.024
F
0.34169
0.38882
2.7020
2.6941
0.9830
−139.512
130.488
See footnote of Table I for explanation of notation.
Table IV
Magnitude and Phase Errors Caused by Introducing(I) Film-Thickness Errors (Δd1 = Δd2 = ±1 nm), (II) Angle-of-incidence Errors (Δδ1 = Δϕ2 = Δϕ3 = ±0.5°) to the QWR Designs of Table III
Error type
ϕ1 (deg)
Magnitude error (|ρn| − 1) × 104
Phase Error (Δn − 90°) (deg)
−
+
−
+
I
52 A
−0.603
0.608
−0.0537
0.0550
B
−0.573
0.576
−0.0546
0.0557
75 A
1.072
−1.067
−0.0817
0.0797
B
1.101
−1.092
−0.0682
0.0660
85 A
7.479
−7.417
−0.1746
0.1609
B
2.988
−2.916
−0.0943
0.0884
C
3.982
−3.949
−0.2133
0.2046
D
−5.518
5.314
−0.3010
0.3106
E
−0.575
0.579
−0.0330
0.0341
F
−0.571
0.575
−0.0333
0.0344
II
52 A
−0.093
0.094
1.9350
−1.9622
B
−0.206
0.213
1.9246
−1.9512
75 A
−2.734
2.850
−5.1858
5.2592
B
1.402
−1.483
−5.0627
5.1303
85 A
−4.169
7.389
−14.0402
15.2408
B
24.356
−26.014
−5.7013
5.7001
C
0.813
2.351
8.0346
−9.1883
D
20.245
−25.010
7.1403
−8.4625
E
0.842
−0.858
6.6376
−6.6682
F
−1.007
1.000
6.5501
−6.5804
Table V
Summary of Design Results for ZnS–Ag Three-Reflection Quarterwave Retarders with s Fast Axis (QWR: |ρn| = 1, Δn = δpn − δsn = −90°) at Three Angles of Incidence and at Wavelength λ = 10.6 μm a
ϕ1
ζ1
ζ2
Dϕ1
Dϕ2
ℛνn
δpn
δsn
52 A
0.60610
0.01252
2.5751
2.4238
0.9520
−119.719
−29.719
B
0.61135
0.47032
2.5751
2.4238
0.9283
74.730
164.730
75 A
0.28732
0.43199
2.6814
2.6207
0.9538
−55.083
34.917
B
0.39292
0.19710
2.6814
2.6207
0.9620
−20.116
69.884
85 A
0.06262
0.46531
2.7020
2.6941
0.9185
−6.017
83.983
B
0.44122
0.96412
2.7020
2.6941
0.9542
28.667
118.667
C
0.93216
0.47129
2.7020
2.6941
0.9010
−16.303
73.697
D
0.96696
0.49627
2.7020
2.6941
0.8580
−108.245
−18.245
E
0.61142
0.61570
2.7020
2.6941
0.9832
139.889
−130.111
F
0.61542
0.61130
2.7020
2.6941
0.9832
139.940
−130.060
See footnote of Table I for explanation of notation.
Table VI
Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1 = Δd2 = ±1 nm),(II) Angle-of-Incidence Errors (Δϕ1 = Δϕ2 = Δϕ3 = ±0.5°) to the QWR Designs of Table V